Key words. database, dynamical system, Conley index, Morse decomposition, Leslie population models, combinatorial dynamics, multiparameter system

Transcription

1 A DATABASE SCHEMA FOR THE ANALYSIS OF GLOBAL DYNAMICS OF MULTIPARAMETER SYSTEMS ZIN ARAI, WILLIAM KALIES, HIROSHI KOKUBU, KONSTANTIN MISCHAIKOW, HIROE OKA, AND PAWE L PILARCZYK Abstract. A generally applicable, automatic method for the efficient computation of a database of global dynamics of a multiparameter dynamical system is introduced. An outer approximation of the dynamics for each subset of the parameter range is computed using rigorous numerical methods and is represented by means of a directed graph. The dynamics is then decomposed into the recurrent and gradient-like parts by fast combinatorial algorithms and is classified via Morse decompositions. These Morse decompositions are compared at adjacent parameter sets via continuation to detect possible changes in the dynamics. The Conley index is used to study the structure of isolated invariant sets associated with the computed Morse decompositions and to detect the existence of certain types of dynamics. The power of the developed method is illustrated with an application to the two-dimensional, density-dependent, Leslie population model. An interactive visualization of the results of computations discussed in the paper can be accessed at the website and the source code of the software used to obtain these results has also been made freely available. Key words. database, dynamical system, Conley index, Morse decomposition, Leslie population models, combinatorial dynamics, multiparameter system AMS subject classifications. Primary: 37B35. Secondary: 37B30, 37M99, 37N25, Introduction. The global dynamics of a nonlinear system can exhibit structures at all spatial scales, for example, the fractal structures associated to chaotic dynamics. The same phenomenon can occur with respect to the parameters, that is, global dynamical structures can change on Cantor sets in parameter space. From the point of view of scientific computation, only a finite amount of information can be computed, and therefore, any computational characterization of global dynamics of a multiparameter system can be expected to represent a dramatic reduction of information. Nevertheless, the computation of global dynamical information is an important problem for applications, which leads to the questions of how to characterize global dynamical structures and how to identify changes in these structures in practice. The PRESTO, Japan Science and Technology Agency / Hokkaido University, Creative Research Initiative Sousei, Sapporo , Japan. Partially supported by Grant-in-Aid for Scientific Research (No ), Ministry of Education, Science, Technology, Culture and Sports, Japan. Department of Mathematical Sciences, Florida Atlantic University, 777 Glades Rd, Boca Raton, FL Partially supported by NSF grant DMS and DOE grant DE-FG02-05ER Department of Mathematics, Kyoto University, Kyoto , Japan. Partially supported by Grant-in-Aid for Scientific Research (No ), Ministry of Education, Science, Technology, Culture and Sports, Japan. Department of Mathematics and BioMaPS, Hill Center-Busch Campus, Rutgers, The State University of New Jersey, 110 Frelinghusen Rd, Piscataway, NJ , USA. Partially supported by NSF grant DMS , by DARPA, and by DOE grant DE-FG02-05ER Department of Applied Mathematics and Informatics, Faculty of Science and Technology, Ryukoku University, Seta, Otsu , Japan. Partially supported by Grant-in-Aid for Scientific Research (No ), Ministry of Education, Science, Technology, Culture and Sports, Japan. Centre of Mathematics, University of Minho, Campus de Gualtar, Braga, Portugal and Department of Mathematics, Kyoto University, Kyoto , Japan. Partially supported by the JSPS Postdoctoral Fellowship No. P06039 and by Grant-in-Aid for Scientific Research (No ), Ministry of Education, Science, Technology, Culture and Sports, Japan. The final version of this preprint has been published in SIAM Journal on Applied Dynamical Systems, Vol. 8, No. 3, 2009, pp , DOI: / , and is available at journal s website. 1

2 2 Z. Arai, W. Kalies, H. Kokubu, K. Mischaikow, H. Oka, and P. Pilarczyk fact that this is a nontrivial task has been made clear by the work of the dynamical systems community over the last century. Identifying and classifying the qualitative properties of models over wide ranges of parameter values is of fundamental importance in many disciplines, and in particular it is of primary interest to many computational biologists [10]. The fact that most topics of interest in systems biology are dynamic in nature suggests the need for a comprehensive, yet efficient method for cataloging the global dynamics of nonlinear systems. In other words, a method is desired which computationally constructs a database of global dynamical behavior of a specific system over a range of parameters. The starting point for our computational methods is Conley s topological approach to dynamics [2], which we review in the next section. The prior work in [1, 5, 9, 12, 24] has shown that Conley theory is an appropriate theoretical base for designing algorithms in computational dynamics. The purpose of this paper is to demonstrate that these ideas can be used to design an efficient computational framework for constructing databases of global dynamics of specific systems over multiple parameters. While the methods we propose are general, we illustrate them using a particular population model which we describe in Section Preliminary ideas. We restrict our attention to the setting of a multiparameter family of dynamical systems given by a continuous function f : X Λ (x, λ) f(x, λ) = f λ (x) X (1.1) where the phase space X is a locally compact metric space and the parameter space Λ is a compact, locally contractible, connected metric space. Note that we do not assume that f λ is a homeomorphism. The fundamental structures in any dynamical system are the invariant sets. A set Z X is invariant at λ Λ if f λ (Z) = Z. Since we cannot perform computations at each parameter value independently, we are interested in considering sets which are invariant with respect to a subset of the parameter space. We use the notation F : X Λ X Λ for the trivial extension of the system to include the parameters as explicit variables defined by F (x, λ) = ( f λ (x), λ ) = ( f(x, λ), λ ). (1.2) For Λ 0 Λ we denote the restriction of F to X Λ 0 by F Λ0 : X Λ 0 X Λ 0. Observe that F = F Λ and that f λ is readily identified with F {λ}. Moreover, given a set S X Λ we denote its restriction to Λ 0 by S Λ0 := S (X Λ 0 ). In particular, we often identify S λ X with S {λ} = S λ {λ}. In this language, a set S X Λ is invariant over Λ 0 if F Λ0 (S Λ0 ) = S Λ0, which is an equivalent way to state that the set S λ is invariant at all λ Λ 0. We wish to identify and characterize a finite collection of invariant sets which determine the global, qualitative behavior of the dynamics. In addition, we need a method for comparing two invariant sets S λ0 and S λ1 at distinct parameters. Deciding what sets to compute and how to compare them are the fundamental issues we attempt to address. We emphasize that while the amount of information that can be computed is finite, this does not mean that we cannot detect or characterize dynamical structures that occur on arbitrarily small scales. For example, topological techniques can be used to identify chaotic dynamics via a semiconjugacy onto a subshift of finite type, see [12]. Moreover, even though we cannot compute the dynamics at each parameter

3 A Database Schema for the Analysis of Global Dynamics of Multiparameter Systems 3 separately, this does not mean that we cannot obtain results which apply to every parameter value. Indeed, the methods we employ do compute mathematically rigorous global structures for invariant sets at all parameter values Model example. To provide perspective on the practicality and utility of our approach, we consider the two-dimensional version of an overcompensatory Leslie population model g : R 2 R 4 R 2 given by [ ] [ ] x1 (θ1 x 1 + θ 2 x 2 )e φ(x1+x2), x 2 px 1 where the fertility rates decay exponentially with population size. This model and its biological relevance is discussed in considerable detail in work of Ugarcovici and Weiss [25]. As indicated there, in view of g(x, θ, p, φ) = φ 1 g(φx, θ, p, 1), the constant φ may be scaled arbitrarily, and thus it suffices to study f : R 2 R 3 R 2 given by (x, λ) = [ x1 x 2 ], θ 1 θ 2 p [ ] (θ1 x f(x, λ) = 1 + θ 2 x 2 )e 0.1(x1+x2) px 1 (1.3) Furthermore, the detailed numerical studies of [25] indicate that this system exhibits a wide variety of different dynamical behavior, which suggests that it provides a meaningful test for the usefulness of the techniques introduced in this paper Outline of the paper. In the remainder of the paper we develop our method and apply it to the Leslie model. In Section 2 we introduce the dynamical structures which we utilize and review Conley theory. In Section 3 we introduce a method for building a combinatorial representation of the dynamics and a means of comparing the computed structures at different parameters. In Section 4 we address the question of efficient computational algorithms based on rectangular grids in X and Λ. In Section 5 we describe a database computed with our methods for the model example introduced in Section 1.2 with p = 0.7 and θ 1, θ 2 varying in selected ranges, and we show how this database can be queried for various dynamical properties. Finally, in Section 6 we briefly mention a result of similar computations where all three parameters are varied. 2. Review of Conley theory. Recall that a compact set N X Λ 0 is an isolating neighborhood for F Λ0 if Inv (N, F Λ0 ) int X Λ0 (N) where Inv (N, F Λ0 ) denotes the maximal invariant set in N under F Λ0, and int X Λ0 (N) denotes the interior of N with respect to the subspace topology on X Λ 0. Isolating neighborhoods are computable and hence provide a means of identifying invariant sets. In particular, an invariant set S Λ0 X Λ 0 is an isolated invariant set if S Λ0 = Inv (N, F Λ0 ) for some isolating neighborhood N. As indicated in [2, 15], the space of isolated invariant sets is a sheaf. More precisely, if N is an isolating neighborhood for F = F Λ and S = Inv (N, F ), then for any Λ 0 Λ, N Λ0 is an isolating neighborhood for F Λ0 and S Λ0 = Inv (N Λ0, F Λ0 ). In particular, for any λ Λ, S {λ} X {λ} is an isolated invariant set for F {λ} : X {λ} X {λ} and S λ X is an isolated invariant set for f λ : X X. To simplify the presentation and analysis, throughout this paper we make use of the following assumption and establish the following notation.

4 4 Z. Arai, W. Kalies, H. Kokubu, K. Mischaikow, H. Oka, and P. Pilarczyk A1: There exists a compact set B X Λ which is an isolating neighborhood for F. Its maximal invariant set is denoted by S := Inv (B, F ). Using the sheaf property of isolated invariant sets, S Λ0 denotes the restriction of S to X Λ 0, which is an isolated invariant set for F Λ0 if Λ 0 is compact. As indicated above, we are interested in understanding the structure of S λ for all λ Λ, and we make use of two essential ideas due to Conley [2]: Morse decompositions and the Conley Index Morse decompositions. Recall that a Morse decomposition of S Λ0 is a finite collection M(S Λ0 ) = {M Λ0 (p) S Λ0 p P Λ0 } of disjoint isolated invariant sets of F Λ0, called Morse sets, which are indexed by the set P Λ0 on which there exists a strict partial order > Λ0, called an admissible order, such that for every (x, λ) S Λ0 \ p P M Λ 0 (p) and any complete orbit γ of F Λ0 through (x, λ) in S Λ0 there exist indices p > Λ0 q such that under F Λ0 ω(γ) M Λ0 (q) and α(γ) M Λ0 (p). Morse decompositions provide a coarse but global description of the dynamics on S Λ0. Remark 2.1. With regard to the construction of a database for global dynamics, we make several important observations concerning Morse decompositions, see [2, 13]. 1. Morse decompositions of S Λ0 are not unique. They can often be refined or coarsened, and many systems have no finest Morse decomposition. 2. The empty set can be a Morse set. 3. Every structure within S Λ0 that is associated with recurrent dynamics, e.g. a fixed point, a periodic orbit, or chaotic dynamics, must lie in some Morse set. Away from the Morse sets the dynamics is gradient-like and the direction of trajectories is captured by an admissible partial order. 4. Given a Morse decomposition, there is a unique minimal admissible partial order called the flow-defined order. Any extension of the flow-defined order which maintains a strict partial order produces an admissible order. 5. Consider a Morse decomposition M(S Λ0 ) = {M Λ0 (p) S Λ0 p P Λ0 } of S Λ0 with admissible order > Λ0. If Λ 1 Λ 0, then the collection of sets {M Λ1 (p) S Λ1 p P Λ0 }, where M Λ1 (p) := M Λ0 (p) (X Λ 1 ), is a Morse decomposition of S Λ1 under F Λ1 with the same admissible order. Observe that since P Λ0 is a strict partially ordered set, a Morse decomposition can be represented as an acyclic directed graph MG(Λ 0 ) called the Morse graph over Λ 0. The elements of the index set P Λ0, which naturally correspond to the Morse sets, are the vertices of the Morse graph over Λ 0, and the edges of the Morse graph over Λ 0 are the minimal order relations which through transitivity generate > Λ0. In other words, for p, q P Λ0 there is a directed edge p q in MG(Λ 0 ) if p > Λ0 q and there does not exist r P Λ0 such that p > Λ0 r > Λ0 q. Computationally, we obtain Morse decompositions indirectly, and often it can be established by further computation that a Morse set is empty. In that case, such

5 A Database Schema for the Analysis of Global Dynamics of Multiparameter Systems 5 a set can be removed from the Morse decomposition as described by the following proposition whose proof follows directly from the definition of a Morse decomposition. Proposition 2.2. Consider a Morse decomposition M(S Λ0 ) = { } M Λ0 (p) S Λ0 p P Λ0 of SΛ0 with admissible order >. Assume M Λ0 (p 0 ) =. Then M (S Λ0 ) := {M Λ0 (p) S Λ0 p P Λ0 \ {p 0 }} is a Morse decomposition of S Λ0 and an admissible order is given by > where for all p, q P Λ0 \ {p 0 } we have p > q if and only if p > q. From Proposition 2.2, if MG(Λ 0 ) is the Morse graph over Λ 0 corresponding to M(S Λ0 ), then MG (Λ 0 ), the Morse graph over Λ 0 corresponding to M (S Λ0 ), has vertices P Λ0 \ {p 0 } and has an edge p q if either p q is an edge in MG(Λ 0 ) or p p 0 q are edges in MG(Λ 0 ). The simplified Morse graph MG (Λ 0 ) is called a trivial reduction of MG(Λ 0 ). Whenever possible, we work with a trivial reduction of a Morse graph. The possibility that a Morse set is trivial can be detected via the Conley index which is described in the next section The Conley index. As explained in Section 3, computational methods exist to find Morse decompositions, see [1]. The Conley index, which is an algebraic topological invariant of isolated invariant sets, is used to understand the structure of the dynamics within a Morse set. To explain the index we begin by considering an arbitrary continuous map g : Z Z on a locally compact metric space and a pair of compact sets N = (N 1, N 0 ) such that N 0 N 1 Z. Consider the pointed quotient space ( N 1 /N 0, [N 0 ] ) obtained by collapsing N 0 to a single point [N 0 ]. Define g N : ( N 1 /N 0, [N 0 ] ) ( N 1 /N 0, [N 0 ] ) by { g(x) if x, g(x) N1 \ N g N (x) = 0 [N 0 ] otherwise. The pair N = (N 1, N 0 ) is an index pair if the map g N is continuous and cl (N 1 \ N 0 ) is an isolating neighborhood [21]. The following two facts about index pairs are most relevant and can be found in [2, 13]. 1. For any isolated invariant set K, there exists at least one index pair N = (N 1, N 0 ) such that K = Inv (cl (N 1 \ N 0 ), g). 2. For any isolated invariant set K, there can exist many index pairs which isolate K. The first fact implies that for any Morse set M Λ0 (p) for the system F Λ0 there exists an index pair N = (N 1, N 0 ) such that the induced map F Λ0,N : ( N 1 /N 0, [N 0 ] ) ( N1 /N 0, [N 0 ] ) is a continuous function and M Λ0 (p) = Inv ( ) cl (N 1 \ N 0 ), F Λ0. Passing to homology leads to a family of group endomorphisms F Λ0,N : H ( N1 /N 0, [N 0 ] ) H ( N1 /N 0, [N 0 ] ). (2.1) The second fact allows for different choices of index pairs which can lead to different group endomorphisms. Thus, to define the Conley index of an isolated invariant set, such as a Morse set M Λ0 (p), one must consider equivalence classes of these group endomorphisms [13]. In constructing our database, we do not utilize the full Conley index; instead, we store a weaker invariant, ( namely the nonzero eigenvalues of F Λ0,N restricted to the torsion-free part of H N1 /N 0, [N 0 ] ). This weaker invariant is chosen because these eigenvalues are readily computed and compared. Remark 2.3. With regard to the construction and interpretation of the database, there are several important observations about the index to be made, see [13].

6 6 Z. Arai, W. Kalies, H. Kokubu, K. Mischaikow, H. Oka, and P. Pilarczyk 1. If the Conley index is trivial, then the map F Λ0,N is nilpotent. This implies ( that the eigenvalues of F Λ0,N restricted to the torsion-free part of H N1 /N 0, [N 0 ] ) are all zero. 2. The empty set is an isolated invariant set with trivial Conley index. Thus, if there ( are nonzero eigenvalues of F Λ0,N restricted to the torsion-free part of H N1 /N 0, [N 0 ] ), then M Λ0 (p). 3. There exist nontrivial isolated invariant sets with trivial Conley index. For example, consider the logistic map g λ (x) = λx(1 x), λ [1, 4], and an index pair N = (N 1, N 0 ) = ( [ 1, 2], { 1, 2} ). The index map F N is nilpotent, but clearly S λ = Inv (N 1 \ N 0, g λ ). 4. The Conley index can be used to reconstruct some of the structure of the dynamics of the associated isolated invariant set. In particular, under appropriate conditions the Conley index can be used to conclude the existence of fixed points, periodic orbits, and chaotic dynamics. 5. Suppose Λ 1 Λ 0 are both contractible. If M Λ0 (p) is a Morse set, or indeed any isolated invariant set, then the nonzero eigenvalues of the torsion-free part of any index map F Λ0,N are identical to the nonzero eigenvalues of the torsion-free part of any index map F Λ1,N for the set M Λ1 (p) = M Λ0 (p) (X Λ 1 ) Conley-Morse graphs. We are now in a position to describe the fundamental element of our database. Let Λ 0 Λ and let M(S Λ0 ) = {M Λ0 (p) S Λ0 p P Λ0 } be a Morse decomposition of S Λ0 with admissible order > Λ0. The Conley-Morse graph over Λ 0 of M(S Λ0 ) is denoted by CMG(Λ 0 ) and consists of MG(Λ 0 ), the Morse graph over Λ 0 with the additional information of the nonzero eigenvalues of the torsion-free part of the index map of each Morse set assigned to the associated node. We present this information by labeling each node in the following format. pk : n { } denotes the fact that the k-th Morse set has nonzero eigenvalues { } on the n-th level of homology. If the k-th Morse set has no nonzero eigenvalues then we write pk : 0. An example of how this information can be used to describe the structure of the dynamics is given in Proposition 5.8. As described in Sections 3 and 4, we compute the Conley-Morse graphs over distinct fixed subregions of parameter space. This raises the question of how to relate the resulting Conley-Morse graphs. One answer is provided in the following definition. Definition 2.4. Assume A1. Let Λ 0, Λ 1 Λ be such that Λ 0,1 := Λ 0 Λ 1 is a nonempty, contractible set. Let M(S Λi ) = {M Λi (p) S Λi p P Λi } for i = 0, 1 be Morse decompositions with admissible orders > i. The associated Morse graphs MG(Λ 0 ) over Λ 0 and MG(Λ 1 ) over Λ 1 are equivalent if there exists an order preserving bijection ι: (P Λ0, > 0 ) (P Λ1, > 1 ) such that M Λ0 (p) (X Λ 0,1 ) = M Λ1 ( ι(p) ) (X Λ0,1 ).

7 A Database Schema for the Analysis of Global Dynamics of Multiparameter Systems 7 The continuation property of the Conley index [2] implies that if MG(Λ 0 ) and MG(Λ 1 ) are equivalent via the order preserving bijection ι: (P Λ0, > 0 ) (P Λ1, > 1 ), then the Conley-Morse graphs CMG(Λ 0 ) and CMG(Λ 1 ) are also equivalent, that is, the nonzero eigenvalues associated to corresponding Morse sets are identical. 3. Combinatorial representation of dynamics. Conley-Morse graphs are the key elements of information stored in our database. In this section we describe a general procedure for computing these graphs. We begin by describing a means of combinatorializing the dynamics. Recall [17] that a grid on a metric space Z is a collection Z of nonempty, compact subsets of Z with the following properties: (a) Z = G Z G, (b) G = cl ( int (G) ) for all G Z, (c) G int (H) = for all G H Z, and (d) if K Z is compact, then {G Z G K } is a finite set. The grid Z has the simple intersection property if G, H Z and G H implies that G H is contractible. The diameter of a grid is defined by diam (Z) := sup G Z diam (G), and the realization map is a function from subsets of Z to subsets of Z defined by A := A A A. Given Y Z, define Z(Y ) := {G Z int (G) Y }. For the remainder of the paper, X and denote grids with the simple intersection property on X and Λ, respectively. Observe that X is a grid for X Λ. As shown in [9], given δ > 0 we can choose grids such that diam (X ) < δ and diam () < δ. A combinatorial multivalued map F : Z Z assigns to each element G Z a finite (possibly empty) subset F(G) of Z. Important for efficient computation is the observation that a combinatorial multivalued map F : Z Z is equivalent to a directed graph with vertices Z and directed edges (G, H) whenever H F(G). A directed graph is closed if each vertex is both the head of at least one edge and the tail of at least one edge. We use combinatorial multivalued maps on the grid X as a means to discretize and combinatorially approximate the dynamics of f. For a more detailed description see [9]. Fix λ Λ and a compact set B λ X. We relate f λ Bλ : B λ X to the combinatorial multivalued map F λ : X (B λ ) X by requiring that F λ outer approximates f λ in the following sense [24]. A multivalued map F λ : X (B λ ) X is called an outer approximation of f λ restricted to X (B λ ) if f λ (G) int ( F λ (G) ) for all G X (B λ ). The property of being an outer approximation is the key to approximating the dynamics of f combinatorially. Since we can only perform a finite number of computations, we cannot compute F λ individually for each λ Λ. Recall that we have made the assumption A1 so that the compact set B X Λ is an isolating neighborhood for F, and S = Inv (B, F ). From the point of view of the computational algorithms it is more convenient to additionally assume the following condition similar in spirit to A1. As it is made clear via Remarks 4.3 and 5.2, these two assumptions are computationally compatible. A1 : For each grid element the set B := X ( λ B λ) has the property that S = Inv ( B, F ). Now for each we consider a multivalued map F : B X with the property that f(g, ) int ( F (G) ) (3.1)

8 8 Z. Arai, W. Kalies, H. Kokubu, K. Mischaikow, H. Oka, and P. Pilarczyk for all G B. Observe that if λ, then F is an outer approximation of f λ restricted to B. We organize the collection of F via the following definition. Set B := ( B {} ) X. A combinatorialization of F on B is the combinatorial multivalued map F : B X defined by F(G, ) = F (G) {}. Note that F is composed of a collection of outer approximations of each F. The following proposition indicates how outer approximations are used to capture invariant sets. Proposition 3.1. Suppose A1 holds and F is a combinatorialization of F on B. Let, and suppose Y B. If N is the maximal subset of Y such that the restriction F : N N is closed, then Inv ( Y, F ) = Inv ( N, F ). Proof: Let (x, λ) Inv ( Y, F ) and choose any G Y that contains x. Let γ x be a complete orbit of F through (x, λ) in Inv ( Y, F ). Since F satisfies (3.1), there exists a sequence {G n } in Y with G 0 = G and G n+1 F (G n ) such that γ x (n) G n for all n Z. Therefore G N, because it lies in a closed subgraph {G n } of Y, which implies that {G n } N, and thus γ x N. Hence (x, λ) Inv ( N, F ). This implies that Inv ( Y, F ) Inv ( N, F ). Since the opposite inclusion is trivial, this concludes the proof. Proposition 3.1 states that the maximal invariant set is captured by the largest closed subgraph of a combinatorialization. We have already commented that grids of arbitrarily small diameter exist in general. If S is the largest closed subgraph of B, then it follows from the results in [9] that S converges to S in Hausdorff metric as the grid diameters of X and tend to zero and the amount of overestimate in the computation of the outer approximation of the image of f by F tends to zero, see Theorem 5.8 and Lemma 7.6 of [9]. We do not provide details here, because even though the convergence results prove that the maximal invariant set can be approximated arbitrarily closely if computations are performed on a sufficiently fine scale, in practice we fix a priori a finest resolution in the phase space and the parameter space with which we do our computations Constructing Conley-Morse graphs. Recall that we have assumed that F satisfies A1. In addition, for the remainder of this section we assume that the set B and the grid X are chosen in such a way that A1 is satisfied. A natural starting point for examining the global structure of both a dynamical system and a directed graph is to look for recurrence. The recurrent set of F : S X is defined by R := {G S there exists a nontrivial path from G to G in S }, where a nontrivial path is any path of non-zero length, including the case a loop from a vertex to itself. The recurrent set R is naturally partitioned into equivalence classes {M (p) p P } called combinatorial Morse sets according to the following equivalence relation:

9 A Database Schema for the Analysis of Global Dynamics of Multiparameter Systems 9 G H if and only if there exists a path in F from G to H and a path in F from H to G. Since every node in S that lies on a cycle is an element of R, we can define a strict partial order on the indexing set P by setting p > q if there exist G M (p), H M (q), and a path from G to H in F. Observe that this construction implies that a combinatorial Morse decomposition can be represented as a directed graph. Let MG(F ) denote the acyclic directed graph with vertices consisting of the elements of P and the minimal set of directed edges p q which generate p > q under transitivity. The following proposition states that given a combinatorial Morse decomposition for an outer approximation F, there is a Morse decomposition of F such that MG(F ) is the Morse graph over for the Morse decomposition. Proposition 3.2. Assume A1 and A1 are satisfied. Let and let {M (p) p P } be the set of combinatorial Morse sets for F. If F ( M (p) ) B for all p P, then the acyclic directed graph MG(F ) which represents the combinatorial Morse sets is a Morse graph over for the Morse decomposition of S defined by M(S ) := {Inv ( M (p), F ) p P }. Moreover, each M (p) is an isolating neighborhood for Inv M (p). Proof: By [9, Theorem 4.1], we know that M (p) is an isolating neighborhood for F, and by [9, Corollary 4.2], M(S ) := {M (p) := Inv ( M (p), F ) p (P, > )} is a Morse decomposition of S. Remark 3.3. Observe that Proposition 3.2 implies that once an appropriate combinatorialization of F has been computed, then for each a Morse graph MG(F ) over is determined which can be associated to a true Morse decomposition M(S ) for F, and this in turn provides a Morse decomposition M(S λ ) for f λ for each λ. The algorithms used to compute the Conley-Morse graphs are discussed in greater detail in Section 4. For the moment we remark that we use the algorithm presented in Section 2.2 of [1] to compute the Morse graph MG(F ). There are a variety of algorithms for determining index pairs, see [8, 18, 19, 24], and in this paper we adopt the approach of [19], see Remark 4.3. For systems defined in R n and simplicial or rectangular grids, there exist algorithms to compute the induced map on homology [8, 14, 20]. Hence for each the Conley-Morse graph CMG(F ) can be determined in a fairly general setting Comparing Conley-Morse graphs. We now turn to the question of comparing the dynamical information over different parameter regions via Conley-Morse graphs. Recall that we assume that the grid has the simple intersection property. Furthermore, we continue to assume that A1 and A1 are satisfied. We also assume that CMG(F ) has been computed for each. We begin with a few definitions. Definition 3.4. To each, there is an associated CMG(F ). Consider 0, 1 such that 0 1. The clutching graph J ( 0, 1 ) is defined to be the bipartite graph with vertices P 0 P 1 (the union of the vertices from MG(F 0 ) and MG(F 1 )) and with an edge (p, q) P 0 P 1 if M 0 (p) M 1 (q).

10 10 Z. Arai, W. Kalies, H. Kokubu, K. Mischaikow, H. Oka, and P. Pilarczyk Observe that if every vertex in P 0 in the clutching graph J ( 0, 1 ) has a unique edge, then we can define the clutching function ι 1, 0 : P 0 P 1 by ι 1, 0 (p) := q for each edge (p, q) of J ( 0, 1 ). Definition 3.5. Consider the set of Conley-Morse graphs over the grid elements of the parameter space, i.e. {CMG(F ) }. Let 0, 1 such that 0 1. If the clutching function ι 1, 0 : P 0 P 1 is defined and gives a directed graph isomorphism from MG(F 0 ) to MG(F 1 ), then we say that the Conley-Morse graphs over 0 and 1, CMG(F 0 ) and CMG(F 1 ), are equivalent. The equivalence classes of {CMG(F ) } with respect to the transitive closure of this relation are called continuation classes. Remark 3.6. We require that ι 1, 0 generate a directed graph isomorphism as opposed to the weaker condition that ι 1, 0 be a bijection, because differences in the partial order may indicate a difference in the dynamics. Proposition 3.7. Assume A1 and A1 are satisfied. Let 0, 1 such that 0 1. If the clutching function ι 1, 0 : P 0 P 1 is a directed graph isomorphism then there exists a Morse decomposition M(S 0 1 ) = {M 0 1 (r) r P 0 1 } with admissible order > 0 1 such that its restriction is the same as the Morse decomposition M(S i ) over i for each i = 0, 1. Specifically, there is a natural correspondence π i : P 0 1 P i such that M i ( πi (r) ) = M 0 1 (r) (X i ) for any r P 0 1, and > 0 1 agrees with > i through the identification. Furthermore, the nonzero eigenvalues associated to the index maps for pairs of corresponding Morse sets M 0 (π 0 (r)) and M 1 (π 1 (r)) are the same. Proof: Define P 0 1 = {(p, q) P 0 P 1 q = ι 1, 0 (p)}, and the natural correspondence π i : P 0 1 P i. Then one can introduce a welldefined partial order > 0 1 on P 0 1 from the isomorphic partial order > i on P i, and π i becomes an order-preserving isomorphism. Now define M 0 1 (r) = M 0 ( π0 (r) ) M 1 ( π1 (r) ). It follows from the construction that the collection {M 0 1 (r) r P 0 1 } forms a Morse decomposition over 0 1. The result now follows from the sheaf property of Morse decompositions and the continuation property of the Conley index. Proposition 3.7 implies that if CMG(F 0 ) and CMG(F 1 ) belong to the same continuation class then there is a path in the parameter space along which the underlying Morse decompositions are related by continuation, and the Conley indices of the corresponding Morse sets are isomorphic. Remark 3.8. It is important to note that belonging to the same continuation class is a weak equivalence relation. In particular, it is possible that λ 0, λ 1

11 A Database Schema for the Analysis of Global Dynamics of Multiparameter Systems 11 and yet S λ0 and S λ1 are not topologically conjugate. Thus, membership in the same continuation class does not imply equivalence on the level of conjugacy. Heuristically, this is because we are studying the dynamics on the level of the grid elements and differences in the dynamics that lie below this scale cannot be observed. Similarly, it is possible that CMG(F 0 ) and CMG(F 1 ) lie in different continuation classes, but the dynamics of f λ0 on S 0 for λ 0 0 and f λ1 on S 1 for λ 1 1 are topologically equivalent. To see that this is the case consider λ i = λ 0 1, but CMG(F 0 ) and CMG(F 1 ) do not belong to the same continuation class. The heuristic explanation for this is that topological conjugacy is independent of the size of the dynamic structures, but the construction of F clearly depends on the size of the grid decomposition. Finally, it is possible to construct examples in which CMG(F 0 ) and CMG(F 1 ) belong to the same continuation class and 0 1, but ι 1, 0 is not a directed graph isomorphism, or even ι 1, 0 may not be defined. This situation can arise from lack of resolution in either the phase space or parameter space. In fact, in practice (see Section 4.6) we employ a local subdivision algorithm that in principle reduces but does not necessarily eliminate the occurrence of phenomenon of this type. The thesis of this paper is that, given grids for the phase space and parameter space, a useful database for the global dynamics is a list of the continuation classes and their relative connectivity. To be more precise we introduce the following notion. Definition 3.9. Assume A1 and A1 are satisfied. The associated continuation graph CG(F) is a graph whose vertices are the continuation classes {( ) } CMG(j), (j) j = 1,..., J where (k) is the set of parameter boxes associated with the k-th continuation class and CMG(k) = CMG(F ) for some (k). Note that all Conley-Morse graphs CMG(F ) over (k) are isomorphic. There is an edge between the j- th and k-th vertices in CG(F) if there exist (j) and (k) such that. The continuation graph is our database. Of course, additional, problem specific information can also be stored. However, as will be indicated in the context of the density dependent Leslie model, this database provides an extremely compressed yet useful means of describing the global dynamics over a broad range of parameter values. We return to these issues when we use the database to investigate the Leslie model. 4. Building the database using rectangular grids. The results of Section 3 indicate how a combinatorialization of F leads to a database. In this section we describe how a combinatorialization can be effectively computed, including details of selected computational aspects of the method. The optimal choice of a suitable grid is determined by the structure of X, Λ, and B X Λ. For many applications Λ R m is a rectangular region, X = R n, and B is also a rectangular region, that is, B = R Λ for some rectangular region R R n. For convenience, we make this assumption, i.e. B λ = R for all λ Λ, throughout the rest of this paper. This leads to the use of a pair of rectangular grids { m [ ζ i = b i + q i, b i + (q i + 1) ζ ] } i q j {0,..., K j 1}, j = 1,..., m (4.1) K i K i i=1 where b, ζ R m and K Z m, { n [ X (d) ξ i = a i + k i 2 d, a i + (k i + 1) ξ ] } i 2 d k j Z, j = 1,..., n i=1 (4.2)

12 12 Z. Arai, W. Kalies, H. Kokubu, K. Mischaikow, H. Oka, and P. Pilarczyk where d Z + and a, ξ R n, and B (d) := X (d) (R) X (d). The choices of K and d define the accuracy of computations, and the choices of a, b, ξ and ζ are determined by the regions one wishes to study. The parameter d allows for the use of an iterative multiscale method described later that is essential for efficient computation of the dynamics Constructing an outer approximation. For each we construct an outer approximation F (d) : B(d) X (d) as follows. For each grid element G B (d), interval arithmetic [16] is used to compute a rectangular box β(g) which contains the image f(g, ). More precisely, the edges of the rectangular grid element G (which are intervals) are inserted directly into the formula for f in place of corresponding variables, and the value of f(g, ) is evaluated using interval arithmetic which provides a rigorous result in the form of a product of intervals. For each G B (d) define F (d) (G) := { H X (d) H β(g) } and observe that f(g, ) int F (d) (d) (G), see Figure 4.1. Thus, F approximation as defined in (3.1). is an outer Fig A rigorous enclosure of the image of each grid box (dark gray) is computed with interval arithmetic (indicated by the dotted line), and then covered by grid boxes (light gray). Having described the construction of F (d) : B(d) X (d), we turn to the need for a multiscale approach. Observe that the number of grid elements in B (d) is 2 nd, and hence the size of the graph of F grows exponentially as a function either of discretization size or dimension of the problem. However, we are only interested in the largest closed subgraph of F on B (d), and thus we only use the restriction of F to S (d), where S(d) covers Inv B which is often a small subset of B Iterative multiscale identification of the invariant set. To avoid unnecessary computation, we use the following iterative multiscale approach motivated by [7]. Let d 0, d 1,..., d l be an increasing sequence of positive integers. Fix. Construct F (d0) : B(d0) X (d0). Following the algorithms in [1] we compute the maximal closed directed graph S (d0) in F (d0). Define Y(d1) B (d1) to be the grid for and S (d0) (d1). Construct F : Y(d1) X (d1). We repeat the construction of S (di) F (di) : Y(di) X (di) until i = l. Note that since we assume A1, Proposition 3.1

13 A Database Schema for the Analysis of Global Dynamics of Multiparameter Systems 13 Fig Computation of S (d i) by means of gradual refinements. In the first row, the lines indicate the subdivision of the region X into 4 n boxes for n = 1,..., 4. No more lines are added in the second row to keep the picture clear. The third row shows a magnified fragment of the region S (d i) (q 1, q 2 ) = (20, 33). with the lines corresponding to those in the second row. Parameter box coordinates: implies that Inv ( S (d l) ) = Inv ( B (d0) ). Example 4.1. To put the above construction into perspective consider the Leslie model (1.3) where we fix p = 0.7. Set Λ := {(θ 1, θ 2 ) [8, 37] [3, 50]}, R := [ 0.001, ] [ 0.001, ] for all λ Λ. The grid in Λ is given by (4.1) with m = 2, b 1 = 8, b 2 = 3, ζ 1 = 29, ζ 2 = 47, and K 1 = K 2 = 50, while the grids X (d) in X = R 2 are given for d = 1,..., 12 by (4.2) with n = 2, a 1 = a 2 = 0.001, ξ 1 = and ξ 2 = Figure 4.2 indicates the evolution of the sets S (d) for d = 1,..., 12, where is given by (q 1, q 2 ) = (20, 33) in (4.1). Remark 4.2. Observe that the diameter of F (d) (G) depends on the diameter of β(g), which in turn is dependent both on the diameter of G and. Currently, we fix the grid and then iteratively refine X (d). Thus, the size of puts an upper bound on the number of iterative steps that lead to a useful refinement of the outer approximation F (d) : Y(d) X (d). A more sophisticated approach would involve an iterative method for both the phase space and the parameter space.

14 14 Z. Arai, W. Kalies, H. Kokubu, K. Mischaikow, H. Oka, and P. Pilarczyk Remark 4.3. By Proposition 3.1, S S (di) for i = 0,..., l. Moreover, is the maximal closed subgraph, and combinatorial Morse sets are closed since S (di) subgraphs, each combinatorial Morse set M (p) for p P is contained in S (di) for each i = 0,..., l. If F ( M (p) ) B for all p P, then by Proposition 3.2, the neighborhoods M (p) isolate Morse sets in a Morse decomposition of S. Moreover, since Morse sets are recurrent, the condition F ( M (p) ) B implies that F ( F (M (p)) \ M (p) ) M (p) =, (4.3) where F : X X is an outer approximation. As described in the next section, this allows us to readily compute the Conley index of Inv ( M ). If the condition F ( M (p) ) B is not satisfied, then we cannot compute the index without extending our computation of F outside of B, and this situation is reported in the output of our computations Conley-Morse graph computation. Given F (d l) : S(d l) X (dl) the algorithm in [1, Section 2.2] is used to compute the Morse graph MG(F ), where F := F (d l). To produce the Conley-Morse graph over requires the computation of the Conley indices of each Morse set. This is done using the algorithms in [14, 20] and the easy observation that if M (p) is a combinatorial Morse set for F and the condition (4.3) is satisfied for F : M (p) F (M (p)) ( X, then M (p) F (M (p)), F (M (p)) \ M (p) ) is a combinatorial index pair as in [19, 20]. In practice, there are memory constraints associated with these algorithms for computing the Conley index. In the computations performed for this paper we did not attempt to compute the Conley index if the index pairs generated contained more than 400,000 boxes. Example 4.4. Returning to Example 4.1, the computed Conley-Morse graph over is indicated in Figure 4.3(a). The three Morse sets M (p i ), i = 2, 4, 5, indicated by the shaded boxes have no nonzero eigenvalues. The numbers in parentheses indicate the number of boxes that define each combinatorial Morse set M (p i ). Remark raises the possibility but does not imply that M (p i ) = for i = 2, 4, 5. This possibility is reinforced by the observation that there are very few boxes in M (p i ) for i = 2, 4, 5. To test whether the Morse sets with trivial index may, in fact, be numerical artifacts we need to be able to study them at a finer level of resolution Combinatorial Morse set refinement. Observe that if M (p) is a combinatorial Morse set determined by an outer approximation F (d), then the restriction F (d) : M (p) M (p) is a closed graph. Choose d greater than d. Define Y (d ) to be the grid for M (p), construct F (d ) : ) Y(d Y (d ), and compute ) S(d, the maximal closed directed graph in F (d ) M (p) is contained in S (d ), because. By Proposition 3.1 all the recurrent dynamics of M (p) = Inv ( S (d ), F ) = Inv ( M (p), F ) Conley-Morse graph reduction test. The first step is to apply the Combinatorial Morse Set Refinement algorithm with d = d l and d = d l + 1. There are two possible outcomes.

15 A Database Schema for the Analysis of Global Dynamics of Multiparameter Systems 15 p0 : 2 {-1} (1) p0 : 2 {-1} (1) p1 : 1 { i, i, 1} (1457) p1 : 1 { i, i, 1} (1457) p4 : 0 (11) p2 : 0 { i, i, 1} (4522) p3 : 0 {1} (105052) p5 : 0 (4) p2 : 0 (8) p6 : 0 {1} (105052) p3 : 0 { i, i, 1} (4522) (a) (b) Fig (a) The Conley-Morse graph computed over the parameter box defined by (q 1, q 2 ) = (20, 33) and d l = 12. The three Morse sets indicated by the shaded boxes have no nonzero eigenvalues. (b) The trivially reduced Conley-Morse graph over. 1. S (d ) =. In this case M (p) =, so we remove the vertex p and replace CMG(F (d l) 2. S (d ) ) with its trivial reduction.. In this case there are two other options: Either we halt and accept CMG(F (d l) ) as the appropriate Conley-Morse graph over, or we apply the Combinatorial Morse Set Refinement procedure to S (d ) process. and repeat the In practice there are two constraints to this procedure. The first is the number of iterations before accepting that a Morse set with trivial index remains in the Conley- Morse graph. In the computations performed for this paper we stopped after three iterations. The second arises from memory considerations. If the combinatorial Morse set consists of a large number of boxes, then representing it on a finer grid can produce an impractically large set. In the computations performed for this paper we did not apply the Combinatorial Morse Set Refinement to sets consisting of more than 40,000 boxes. Example 4.5. Continuing with Example 4.4, after applying one iteration of the Conley-Morse Graph Reduction Test with d = 13 it is determined that M (p k ) =, for k = 2, 4, 5. This leads to the trivially reduced Conley-Morse graph indicated in Figure 4.3(b). The sets M (p k ) for the reduced Conley-Morse graph over are indicated in Figure 4.4. Recall that the desired database is a continuation graph. The first step is to create the continuation classes. Observe that over each we have computed the trivially reduced Conley-Morse graphs CMG(F ). We still retain the information of the boxes {M (p) p P } which define the vertices of CMG(F ) and of course the ordering > between the vertices Continuation graph construction. Let 0 and 1 be grid elements such that 0 1. In order to determine whether CMG(F 0 ) and CMG(F 1 ) are equivalent, we proceed with the following steps: 1. If the cardinalities of P 0 and P 1 differ then 0 and 1 do not belong to

16 16 Z. Arai, W. Kalies, H. Kokubu, K. Mischaikow, H. Oka, and P. Pilarczyk Fig The sets M (p i ), i = 0, 1, 2, 3, for the reduced Conley-Morse graph from Figure 4.3(b). The color coding in the two figures match, thus the green and red regions indicate attracting neighborhoods for f λ for all λ in the parameter box defined by (q 1, q 2 ) = (20, 33) and d l = 12. The set M (p 0 ) covers the origin and consists of a single box at the lower left corner of the picture and is barely visible at this resolution. the same continuation class. 2. If the cardinalities of P 0 and P 1 agree, then we construct the clutching graph (see comment preceding Definition 3.5). If the clutching graph defines a directed graph isomorphism between the Morse graphs MG(F 0 ) and MG(F 1 ), then the Conley-Morse graphs CMG(F 0 ) and CMG(F 1 ) over 0 and 1 belong to the same continuation class. 3. The clutching graph can fail to define a directed graph isomorphism between the Morse graphs MG(F 0 ) and MG(F 1 ) in several ways: The partial orders do not agree, in which case CMG(F 0 ) and CMG(F 1 ) are not identified as belonging to the same continuation class. There is a vertex with no edge in the clutching graph, in which case CMG(F 0 ) and CMG(F 1 ) do not belong to the same continuation class. There is a vertex with two or more edges in the clutching graph. Recall that the edge (p, q) is in the clutching graph if M 0 (p) M 1 (q). Assume the edges at the vertex p are given by {(p, q i ) i = 1,..., I}. Then we apply several iterations of the Combinatorial Morse Set Refinement algorithm to M 0 (p) and M 1 (q i ), i = 1,..., I. If after performing this at each vertex for which there are multiple edges, the clutching graph reduces to a directed graph isomorphism, then CMG(F 0 ) and

17 A Database Schema for the Analysis of Global Dynamics of Multiparameter Systems 17 CMG(F 1 ) belong to the same continuation class; otherwise CMG(F 0 ) and CMG(F 1 ) belong to different continuation classes. This procedure applied to all pairs of adjacent boxes in the parameter space results in the set of continuation classes and the continuation graph which defines the database having been determined. Note that although this procedure admits the situation in which two Conley-Morse graphs are identified as belonging to the same continuation class even if the clutching function does not define a directed graph isomorphism, in such a case the conclusion of Proposition 3.7 still holds true, because the clutching function of the refined combinatorial Morse decompositions does provide the isomorphism. 5. Database for the Leslie model with p = 0.7. In this section we present the results of the computational procedure described in Section 4 applied to the density dependent Leslie model (1.3) when p = 0.7. The procedure was implemented in an efficient program written in C++. An interactive presentation of the results of computations which we did can be found at and the source code of the general purpose software used to compute this database has also been made freely available. Observe that fixing p = 0.7 results in a two-dimensional parameter space. We compute the continuation graph over the parameter space Λ := {(θ 1, θ 2 ) [8, 37] [3, 50]}. We choose an equipartitioned grid for this parameter space given by (4.1) with m = 2, b 1 = 8, b 2 = 3, ζ 1 = 29, ζ 2 = 47, and K 1 = K 2 = 50. Recasting the Leslie model in the general form of (1.2) and recalling that x 1 and x 2 represent population sizes, we are interested in studying F : (R + ) 2 Λ (R + ) 2 Λ where R + := [0, ). There are two essential observations. Remark 5.1. For λ = (θ 1, θ 2 ), define B λ := { (x 1, x 2 ) (R + ) 2 0 x 1 10(θ 1 + θ 2 )e 1, 0 x 2 10p(θ 1 + θ 2 )e 1}. A direct calculation shows that F 2( (R + ) 2 Λ ) B := λ Λ B λ {λ}. In particular, any invariant set for the Leslie model on (R + ) 2 must lie in B. Furthermore, F (0, λ) = (0, λ) and {0} (0, ) f λ ( (R + ) 2). Remark 5.2. There are some technical reasons why we do not compute on the set B (R + ) 2 Λ as defined in Remark 5.1. First, F (0, λ) = (0, λ) so that B is not an isolating neighborhood as a subset of R 2 Λ. While this would not violate A1 restricted to (R + ) 2 Λ, it does imply that the Conley index of the invariant set {0} computed restricted to B would not be capable of measuring the hyperbolicity of the origin. Our approach to this problem is pragmatic, we can explicitly compute the eigenvalues at the origin as µ ± = θ 1 ± θ pθ 2. 2 Observe that restricted to Λ, µ + > 1. This information can then be used to compute the index analytically. However, this is not the only reason not to compute on B. As exhibited in the figures in the next section, the maximal invariant set in B comes very close to the coordinate axes, even though it does not touch the axes by

18 18 Z. Arai, W. Kalies, H. Kokubu, K. Mischaikow, H. Oka, and P. Pilarczyk Remark 5.1. This causes a problem again in isolating and computing the Conley index even for some Morse sets that do not include the origin, when the computations are restricted to the first quadrant. Given these issues, we have chosen to compute on a larger rectangle R := [ 0.001, ] [ 0.001, ] in R 2 and consider the origin analytically. With respect to the dynamics on R 2, the origin undergoes a period doubling bifurcation since µ < 1 if θ 2 > 1 p θ and 1 < µ < 0 if θ 2 < 1 p θ The eigenvector v + associated with µ + can be chosen to lie in the positive orthant. The eigenvector associated with µ lies outside the positive cone. Thus, this period doubling bifurcation has no direct impact on the ordering of the Morse decomposition of S λ := Inv (R, f λ ). Combining this with the observation that µ + > 1, one can conclude that for any λ Λ the origin is always a repeller for S λ. These observations imply that, even though the condition A1 does not hold near the period-doubling bifurcation, this bifurcation plays no role in the construction of the continuation classes. Moreover, the Conley index of the origin provides no information, and hence can be ignored in the database. Using the grids defined in Example 4.1, we compute the continuation graph using the iterative method in Section 4.2 on the grids B (d) for d = 6,..., 12. The output is indicated in Figure 5.1. Since there is no natural order on the continuation classes, we have labeled them from Class 1 to Class 17, according to their volume in parameter space, beginning with the largest region. Class 17 [1 box] Class 16 [1 box] Class 15 [1 box] Class 10 [43 boxes] Class 1 [890 boxes] Class 2 [759 boxes] Class 7 [66 boxes] Class 3 [251 boxes] Class 6 [73 boxes] Class 9 [50 boxes] Class 4 [196 boxes] Class 8 [65 boxes] Class 13 [1 box] Class 14 [1 box] Class 5 [88 boxes] Class 11 [12 boxes] Class 12 [2 boxes] Fig The continuation graph computed for the density dependent Leslie population model (1.3) with Λ = {θ = (θ 1, θ 2 ) [8, 37] [3, 50]}, and p = 0.7. The label of each node indicates the class number and the number of boxes in (j), and the border color agrees with Figure 5.2. The advantage of the continuation graph is that it is dimension independent, and thus it can be used to organize the information about the dynamics independent of the dimension of the parameter space. In this example, however, the parameter space is two-dimensional, thus we also present a continuation diagram in Figure 5.2 where each continuation class with more than one parameter box is identified by some color. Thus, it is easy to see the sets (j) which define the parameter values for which the Conley-Morse graphs are valid. The Conley-Morse graphs associated to the continuation classes, which are the basic items of the database, are shown in the next subsection.

19 A Database Schema for the Analysis of Global Dynamics of Multiparameter Systems 19 Fig Continuation diagram computed for the two-dimensional Leslie population model with θ 1 [8, 37], θ 2 [3, 50], p = 0.7. Grid sizes: in the parameter space for (θ 1, θ 2 ), and in the phase space [ 0.001, ] [ 0.001, ] Catalog of continuation classes. This section contains a list of Conley- Morse graphs and Morse decompositions at selected parameter boxes for all the continuation classes found for the two-dimensional Leslie model with two varying parameters, as discussed above. Recall that each node in a Conley-Morse graph is labeled pk : n { } where k labels the Morse set, { } indicates the nonzero eigenvalues, and n indicates the level of homology of the index map on which these eigenvalues arise. If the k-th Morse set has no nonzero eigenvalues then we write pk : 0. Since, as indicated in Remark 5.2, the origin is an exceptional Morse set, we indicate it with a shaded box and do not include any index information. The boxes are color coded to match the combinatorial Morse sets that are shown in the right panel. We do not consider the figures showing the combinatorial Morse sets to be part of the database for the following reasons: 1. The boxes that make up the combinatorial Morse sets will, in general, be different for each. 2. For higher dimensional problems visualizing the combinatorial Morse sets becomes impractical if not impossible. 3. Storing the combinatorial Morse sets becomes prohibitive for higher dimensional problems. 4. We do not have a method for directly querying the combinatorial Morse sets.

20 20 Z. Arai, W. Kalies, H. Kokubu, K. Mischaikow, H. Oka, and P. Pilarczyk p2 : origin p1 : 2 {1} p0 : 1 {1}, 0 {1} 1. The Conley-Morse graph CMG(1) and the sets M(p) at the box (35, 32). p1 : origin p0 : 0 {1} 2. The Conley-Morse graph CMG(2) and the sets M(p) at the box (10, 13). p2 : origin p1 : 1 { i, i} p0 : 0 { i, i, 1} 3. The Conley-Morse graph CMG(3) and the sets M(p) at the box (18, 12).

21 A Database Schema for the Analysis of Global Dynamics of Multiparameter Systems 21 p3 : origin p2 : 1 { i, i, 1} p1 : 0 { i, i, 1} p0 : 0 {1} 4. The Conley-Morse graph CMG(4) and the sets M(p) at the box (20, 22). p2 : origin p1 : 1 {-1} p0 : 0 {-1, 1} 5. The Conley-Morse graph CMG(5) and the sets M(p) at the box (0, 45). p2 : origin p1 : 0 p0 : 0 {1} 6. The Conley-Morse graph CMG(6) and the sets M(p) at the box (11, 11).

22 22 Z. Arai, W. Kalies, H. Kokubu, K. Mischaikow, H. Oka, and P. Pilarczyk p3 : origin p2 : 2 {1} p1 : 1 { i, i, 1} p0 : 0 { i, i, 1} 7. The Conley-Morse graph CMG(7) and the sets M(p) at the box (31, 22). p2 : origin p1 : 0 p0 : 0 {1} 8. The Conley-Morse graph CMG(8) and the sets M(p) at the box (27, 39). p1 : origin p0 : 0 {1} 9. The Conley-Morse graph CMG(9) and the sets M(p) at the box (31, 47).

23 A Database Schema for the Analysis of Global Dynamics of Multiparameter Systems 23 p1 : 2 {1} p0 : NO ISOLATION 10. The Conley-Morse graph CMG(10) and the sets M(p) at the box (47, 2). p3 : origin p2 : 0 p1 : 0 p0 : 0 {1} 11. The Conley-Morse graph CMG(11) and the sets M(p) at the box (22, 47). p4 : origin p3 : 1 { i, i, 1} p1 : 0 p2 : 0 { i, i, 1} p0 : 0 {1} 12. The Conley-Morse graph CMG(12) and the sets M(p) at the box (23, 49).

24 24 Z. Arai, W. Kalies, H. Kokubu, K. Mischaikow, H. Oka, and P. Pilarczyk p4 : origin p3 : 1 { i, i, 1} p1 : 0 p2 : 0 { i, i, 1} p0 : 0 {1} 13. The Conley-Morse graph CMG(13) and the sets M(p) at the box (22, 45). p2 : origin p1 : 0 p0 : 0 {1} 14. The Conley-Morse graph CMG(14) and the sets M(p) at the box (26, 0). p3 : origin p1 : 0 p2 : 2 {1} p0 : 1 {1}, 0 {1} 15. The Conley-Morse graph CMG(15) and the sets M(p) at the box (27, 0).